Out of all the letters of the Greek alphabet π enjoys the greatest fame, being forever associated with geometry. Take a circle, any circle, and you will find that the ratio between its circumference and its diameter is equal to π. Similarly, the ratio between its area and its radius squared is equal to π.
The fact that these ratios are the same for any circle has been known for a long time. A Babylonian clay tablet from between 1900 and 1600 BC puts the value of this ratio at 3.125, the Egyptian Rhind papyrus from about 1650 BC puts it at 3.1605 and the Indian Shukba Sutras, from around 600 BC, put it at 3.088.
But we now know that π is far more awkward than the ancients appreciated. It’s an irrational number, meaning that it cannot be written as a fraction. Irrational numbers were highly unpopular with the Pythagoreans who believed that whole numbers and their ratios, the fractions, hold the key to the Universe. Legend has it that Hippasus of Metapontum (5th century BC) was thrown over board a ship for discovering them.
π was finally proved to be irrational in 1761 but only three decades later it was found to have more awkwardness up its sleeve. Other irrational numbers, such as Ö2, have the nice property that they are solutions to simple equations, x2=2 in the case of Ö2. π isn’t the solution of any such equation. To be precise, it is not the solution of any equation of the form
anxn+ an-1xn-1+ … +a1x+ a0=0
where a0, a1, and so on, up to an, are whole numbers. This makes π what is called a transcendental number, and one of only a few numbers to have been proven as such. Its transcendentality hampers another problem beloved by the ancients: to construct a square that has the same area as a given circle using only ruler and compass. Because π is transcendental squaring the circle in such a way is in fact impossible.
π is also intimately bound up with infinity: being irrational means that its decimal expansion is infinite. Whenever you’re looking at a representation of π as something like 3.141 or 3.14159265 (which is what you see on your Cube) you’re looking at an approximation. Not only is the expansion infinite but it doesn’t repeat: no matter how far you go along it, you’ll never find that the rest of it is made up of a repeating block of numbers.